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Path Integrals in Polymer Physics H. Kleinert, PATH INTEGRALS January 6, 2016 (/home/kleinert/kleinert/books/pathis/pthic15.tex)Thou com’st in such a questionable shape, That I will speak to thee. William Shakespeare (1564–1616), Hamlet 15 Path Integrals in Polymer Physics The use of path integrals is not confined to the quantum-mechanical description point particles in spacetime. An important field of applications lies in polymer physics where they are an ideal tool for studying the statistical fluctuations of line- like physical objects. 15.1 Polymers and Ideal Random Chains A polymer is a long chain of many identical molecules connected with each other at joints which allow for spatial rotations. A large class of polymers behaves ap- proximately like an idealized random chain. This is defined as a chain of N links of a fixed length a, whose rotational angles occur all with equal probability (see Fig. 15.1). The probability distribution of the end-to-end distance vector xb xa of such an object is given by − N 1 N P (x x )= d3∆x δ( ∆x a) δ(3)(x x ∆x ). (15.1) N b − a n 4πa2 | n| − b − a − n nY=1 Z nX=1 The last δ-function makes sure that the vectors ∆xn of the chain elements add up correctly to the distance vector x x . The δ-functions under the product enforce b − a the fixed length of the chain elements. The length a is also called the bond length of the random chain. The angular probabilities of the links are spherically symmetric. The factors 1/4πa2 ensure the proper normalization of the individual one-link probabilities 1 P (∆x)= δ( ∆x a) (15.2) 1 4πa2 | | − in the integral 3 d xb P1(xb xa)=1. (15.3) Z − The same normalization holds for each N: 3 d xb PN (xb xa)=1. (15.4) Z − 1031 1032 15 Path Integrals in Polymer Physics Figure 15.1 Random chain consisting of N links ∆xn of length a connecting xa = x0 and xb = xN . If the second δ-function in (15.1) is Fourier-decomposed as N 3 d k N (3) ik(xb−xa)−ik ∆xn δ xb xa ∆xn = e n=1 , (15.5) − − ! (2π)3 nX=1 Z P we see that P (x x ) has the Fourier representation N b − a d3k x x ik(xb−xa) ˜ k PN ( b a)= 3 e PN ( ), (15.6) − Z (2π) with N 1 P˜ (k)= d3∆x δ( ∆x a)e−ik∆xn . (15.7) N n 4πa2 | n| − nY=1 Z Thus, the Fourier transform P˜N (k) factorizes into a product of N Fourier- transformed one-link probabilities: N P˜N (k)= P˜1(k) . (15.8) h i These are easily calculated: 1 sin ka ˜ k 3 x x −ik∆x P1( )= d ∆ 2 δ( ∆ a)e = . (15.9) Z 4πa | | − ka The desired end-to-end probability distribution is then given by the integral d3k R ˜ k N ikR PN ( ) = 3 [P1( )] e Z (2π) 1 ∞ sin ka N = 2 dkk sin kR , (15.10) 2π R Z0 " ka # H. Kleinert, PATH INTEGRALS 15.2 Moments of End-to-End Distribution 1033 where we have introduced the end-to-end distance vector R x x . (15.11) ≡ b − a The generalization of the one-link distribution to D dimensions is 1 x x P1(∆ )= D−1 δ( ∆ a), (15.12) SDa | | − with SD being the surface of a unit sphere in D dimensions [see Eq. (1.558)]. To calculate 1 ˜ k D x x −ik∆x P1( )= d ∆ D−1 δ( ∆ a)e , (15.13) Z SDa | | − we insert for e−ik∆x = e−ik|∆x| cos ∆ϑ the expansion (8.130) with (8.101), and use Y0,0(xˆ)=1/√SD to perform the integral as in (8.250). Using the relation between 1 modified and ordinary Bessel functions Jν(z) −iπ/2 −iπ/2 Iν (e z)= e Jν(z), (15.14) this gives Γ(D/2) P˜ (k)= J (ka), (15.15) 1 (ka/2)D/2−1 D/2−1 where Jµ(z) is the Bessel function. 15.2 Moments of End-to-End Distribution The end-to-end distribution of a random chain is, of course, invariant under rotations so that the Fourier-transformed probability can only have even Taylor expansion coefficients: ∞ (ka)2l P˜ (k) = [P˜ (k)]N = P . (15.16) N 1 N,2l (2l)! Xl=0 The expansion coefficients provide us with a direct measure for the even moments of the end-to-end distribution. These are defined by 2l D 2l R d R R PN (R). (15.17) h i ≡ Z 2l The relation between R and PN,2l is found by expanding the exponential under the inverse of the Fourierh integrali (15.6): ∞ ( ikR)n P˜ (k)= dDR e−ikRP (R)= dDR − P (R), (15.18) N N n! N Z nX=0 Z 1I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 8.406.1. 1034 15 Path Integrals in Polymer Physics and observing that the angular average of (kR)n is related to the average of Rn in D dimensions by 0 , n = odd, (kR)n = kn Rn (n 1)!!(D 2)!! (15.19) h i h i − − , n = even. (D + n 2)!! − The three-dimensional result 1/(n+1) follows immediately from the angular average 1 n (1/2) −1 d cos θ cos θ being 1/(n + 1) for even n. In D dimensions, it is most easily derivedR by assuming, for a moment, that the vectors R have a Gaussian distribution (0) 2 3/2 −R2D/2Na2 PN (R)=(D/2πNa ) e . Then the expectation values of all products of Ri can be expressed in terms of the pair expectation value 1 R R (0) = δ a2N (15.20) h i ji D ij via Wick’s rule (3.305). The result is 1 R R R (0) = δ anN n/2, (15.21) h i1 i2 ··· in i Dn/2 i1i2i3...in with the contraction tensor δi1i2i3...in of Eqs. (8.64) and (13.89), which has the re- cursive definition δi1i2i3...in = δi1i2 δi3i4...in + δi1i3 δi2i4...in + ...δi1in δi2i3...in−1 . (15.22) A full contraction of the indices gives, for even n, the Gaussian expectation values: (D + n 2)!! Γ(D/2+ n/2) 2n/2 Rn (0) = − anN n/2 = anN n/2, (15.23) h i (D 2)!!Dn/2 Γ(D/2) Dn/2 − for instance (0) (D + 2) (0) (D +2)(D + 4) R4 = a4N 2, R6 = a6N 3. (15.24) D D2 D E D E By contracting (15.21) with k k k we find i1 i2 ··· in 1 (kR)n (0) =(n 1)!! (ka)nN n/2 = kn (R)n (0) d , (15.25) h i − Dn/2 h i n with (n 1)!!(D 2)!! d = − − . (15.26) n (D + n 2)!! − Relation (15.25) holds for any rotation-invariant size distribution of R, in particu- lar for PN (R), thus proving Eq. (15.17) for random chains. Hence, the expansion coefficients P are related to the moments R2l by N,2l h iN P =( 1)ld R2l , (15.27) N,2l − 2lh i H. Kleinert, PATH INTEGRALS 15.2 Moments of End-to-End Distribution 1035 and the moment expansion (15.16) becomes ∞ ( 1)l(k)2l P˜ (k)= − d R2l . (15.28) N (2l)! 2l h i Xl=0 Let us calculate the even moments R2l of the polymer distribution P (R) h i N explicitly for D = 3. We expand the logarithm of the Fourier transform P˜N (k) as follows: ∞ 2l l sin ka 2 ( 1) B2l 2l log P˜N (k)= N log P˜1(k)= N log = N − (ka) , (15.29) ka ! (2l)!2l Xl=1 where B are the Bernoulli numbers B =1/6, B = 1/30,... Then we note that l 2 4 − for a Taylor series of an arbitrary function y(x) ∞ a y(x)= n xn, (15.30) n! nX=1 the exponential function ey(x) has the expansion ∞ b ey(x) = n xn, (15.31) n! nX=1 with the coefficients b n 1 a mi n = i . (15.32) n! mi! i! {Xmi} iY=0 The sum over the powers mi =0, 1, 2,... obeys the constraint n n = i m . (15.33) · i Xi=1 Note that the expansion coefficients an of y(x) are the cumulants of the expansion y(x) coefficients bn of e as defined in Section 3.17. For the coefficients an of the expansion (15.29), 2l l N2 ( 1) B2l/2l for n =2l, an = − − (15.34) ( 0 for n =2l +1, we find, via the relation (15.27), the moments l 1 N22i( 1)iB mi R2l = a2l( 1)l(2l + 1)! − 2i , (15.35) h i − mi! " (2i)!2i # {Xmi} iY=1 with the sum over mi =0, 1, 2,... constrained by 1036 15 Path Integrals in Polymer Physics l l = i m . (15.36) · i Xi=1 For l = 1 and 2 we obtain the moments 5 2 R2 = a2N, R4 = a4N 2 1 . (15.37) h i h i 3 − 5N In the limit of large N, the leading behavior of the moments is the same as in (15.20) and (15.24). The linear growth of R2 with the number of links N is characteristic for a random chain. In the presenceh ofi interactions, there will be a different power behavior expressed as a so-called scaling law R2 a2N 2ν .
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